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Exercise 3, page 13 from Golan's book ("The Linear Algebra a Beginning Graduate Student Ought to Know"): Define a new operation $\circ$ on $\mathbb{R}$ by setting $a\circ b= a^{3}b.$ Show that $\mathbb{R}$, on which we have the usual addition and this new operation as multiplication, satisfies all of the axioms of a field with the exception of one.

My work: The new operation is not commutative and there is no identity of multiplication. If there exists $x\in \mathbb{R}$ such that $a\circ x= a = x\circ a$, then $a^{3}x=a=x^{3}a$. If $a\neq 0$ then $x=1$ and the first equality would gives us $a=\pm 1$.

Is the exercise wrong or I am doing something wrong?

Thanks in advance!

PS.Please, correct any mistake I made.

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Certainly $1$ is a left identity. I don't know what book you're referring to, but it might only require left identity (which together with commutativity gives right identity. This is silly though). – Qiaochu Yuan Jan 25 '12 at 21:14
There are many different ways of specifying the axioms of a field, so it is difficult to say. One could state "multiplication is commutative" as one axiom, and then only require multiplicative identities and inverses on one side in the other axioms (just like we can define a group with axioms that explicitly require the identity and the inverses to be two-sided, or we can define a group with axioms that only require a left identity and left inverses). So you'd need to look at the axioms very carefully to see how they are stated, in order to decide which one(s) are not satisfied. – Arturo Magidin Jan 25 '12 at 21:15
Can you post the title of the textbook in addition to the author? I found this: "The Linear Algebra a Beginning Graduate Student Ought to Know"; is this the one? – Srivatsan Jan 25 '12 at 21:20
Hello Srivatsan! That is the book! I will give the reference next time. Thanks for point it out. – spohreis Jan 25 '12 at 21:26
Note that the axiom on page 6 does not require the identity to be 2-sided. But it does require a right identity, and that doesn't hold here either. So you may have had a minor technical oversight, but you're still correct that there is more than one property missing. – Jonas Meyer Jan 25 '12 at 22:13

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